广义特征值问题GEVP高层建筑结构动力学算例和Matlab程序
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m4 k4 / 2 m3 k3 / 2 m2 k2 / 2 m1 k1 / 2 k1 / 2 k3 / 2 k /2 k2 / 2 k4 / 2
x4 (t )ቤተ መጻሕፍቲ ባይዱ
x3 (t )
x2 (t )
x1 (t )
Answer: a) Determine the governing equations for the system. From the analysis of free-body diagrams, we write
2
) c 4 u 4 sin( 4 t
2
)
(6)
Solving a set of linear simultaneous equations
u1
u2
u3
u 4 c1
c2
c3
we have
c1
c2
c3
c 4 - 0.0414 0.0508 0.0130 - 0.0063
b)
T
If the stiffness matrix of the system is known, how would you determine the natural frequencies of the system without having to solve a generalized eigenvalue problem? What is the benefit of determining the natural frequencies using this particular approach? From equation
which is, in matrix form,
m1 0 0 0
0 m2 0 0
0 0 m3 0
0 k1 k 2 k 0 2 x 0 0 m4 0
1
k2 k2 k3 k3 0
0 k3 k3 k 4 k4
0 0 0 0 0 4000 10000 5000 0 4000 0 0 5000 10000 5000 0 x 0 x 0 0 0 4000 0 5000 10000 5000 0 0 4000 0 5000 5000 0 0
T
orthogonality of the eigenvectors, we have
T u1 Mu 3 0 T u2 Mu 3 0 T u3 Mu 3 1
Solving the above equations (Appendix B), we have
u 3 0.6591 - 0.4950 0.1588
e) By definition, a node of a mode shape is simply the location of zero displacement. Determine the node locations for modes 2 to 4. Under what conditions may the node locations be important? Please give at least 2 examples. Linear interpolation of the curves of modes 2 to 4 gives the three nodes locations
3
Answer: a) Determine the third normalized eigenvector of the system Assume the third normalized eigenvector of the system to be u 3 u 31 , u 32 , u 33 , then, from the
mode Node location ( ith story) 2 3.0 3 1.7423, 3.5740 4 1.3949, 2.5321, 3.7169
If we want to excite the structure to measure the vibration and do some modal analyses, it’s important to select the proper nodes for excitation. If the point for excitation happens to be at or near a node of a mode, the specific mode will not be excited and can not be measured. A second example to illustrate the importance of the locations of nodes is related to joint two large machine parts or components together. If the points to link (assembly) the two elements together coincide with the nodes of the principle modes, the bolting force will vary little due to vibration of the structures.
Λ diag (0.1508,1.2500, 2.9341,4.4151) , ω diag (0.3883,1.1180,1.7129,2.1012)
- 0.1140 - 0.2143 u4 ] - 0.2887 - 0.3283 0.2887 0.3283 0.2887 - 0.1140 0.0000 - 0.2887 - 0.2887 0.2143 - 0.2143 0.3283 - 0.2887 0.1140
4
numerically efficient algorithms have been developed over the years specifically for the solution of a SEVP. Second, the behavior of a SEVP is much better understood, and many theorems have been formulated for a SEVP as opposed to a GEVP. Prove the orthogonality conditions of the mode shapes for the SEVP. [Hint: you may want to consider the adjoint eigenvalue problem.] Prove: If [A] is symmetric, we can prove the orthogonality conditions of the mode shapes for the SEVP. However, if [A] is nonsymmetric, we should prove the biorthogonality conditions instead. Actually the orthogonality condition can be derived easily if the biorthogonality conditions are proved. a) biorthogonality conditions For the eigenvalue problem
Simplified, we obtain
(2)
4 0 0 0
0 4 0 0
0 0 4 0
0 0 10 5 0 5 10 5 0 0 x 0 x 0 5 10 5 0 4 0 5 5 0
(3)
Solving the above GEVP with a MATLAB script program (Appendix A), we obtain the 4 modes of the 4-story building
T
(7)
Thus, the free response of the system due to the intial displacement is as expressed inn Eq.(6), where
i and ci , (i 1,2,3,4) are given by Eqs.(4) and (7) respectively.
Ku 2 Mu
we know
i
( Ku i )1 , i 1,2, ( Mu i )1
By this approach, it avoids the time-consuming computation to solve a generalized eigenvalue problem and can get the result faster and more economically. 3. In class, we considered the following symmetric generalized eigenvalue problem (GEVP) [K]u = [M]u and proved the orthogonality conditions of the mode shapes with respect to the mass and stiffness matrices. Often it is more desirable to solve a standard eigenvalue problem (SEVP) of the form [A]u = u where [A] is generally nonsymmetric. The rationale for solving the SEVP is two-fold. First, many
1 (k1 k 2 ) x1 k 2 x 2 0 m1 x 2 k 2 x1 (k 2 k 3 ) x 2 k 3 x3 0 m2 x , 3 k 3 x 2 (k 3 k 4 ) x3 k 4 x 4 0 m3 x 4 k 4 x3 k 4 x 4 0 m4 x
x ci u i sin( i t i )
i 1
4
Due to the initial displacement, we have
x c1u1 sin(1t
2
) c 2 u 2 sin( 2 t
c 4 x0
T
2
) c3 u 3 sin( 3t
(4)
U [u1
u2
u3
(5)
c) Plot of the mode shapes. The mode shapes can be plotted by the MATLAB script program (Appendix A) as well
2
d) Determine the free response of the system due to the initial displacement. Under arbitrary initial conditions, the response of the system can be expressed as
0 0 x 0 k4 k4
(1)
b) Determine the modes of vibration. Assume mi = 4000 kg and ki = 5000 N/m, for i = 1, . . . , 4. For the assumed values of mi and ki, we have, from Eq.(1),
x4 (t )ቤተ መጻሕፍቲ ባይዱ
x3 (t )
x2 (t )
x1 (t )
Answer: a) Determine the governing equations for the system. From the analysis of free-body diagrams, we write
2
) c 4 u 4 sin( 4 t
2
)
(6)
Solving a set of linear simultaneous equations
u1
u2
u3
u 4 c1
c2
c3
we have
c1
c2
c3
c 4 - 0.0414 0.0508 0.0130 - 0.0063
b)
T
If the stiffness matrix of the system is known, how would you determine the natural frequencies of the system without having to solve a generalized eigenvalue problem? What is the benefit of determining the natural frequencies using this particular approach? From equation
which is, in matrix form,
m1 0 0 0
0 m2 0 0
0 0 m3 0
0 k1 k 2 k 0 2 x 0 0 m4 0
1
k2 k2 k3 k3 0
0 k3 k3 k 4 k4
0 0 0 0 0 4000 10000 5000 0 4000 0 0 5000 10000 5000 0 x 0 x 0 0 0 4000 0 5000 10000 5000 0 0 4000 0 5000 5000 0 0
T
orthogonality of the eigenvectors, we have
T u1 Mu 3 0 T u2 Mu 3 0 T u3 Mu 3 1
Solving the above equations (Appendix B), we have
u 3 0.6591 - 0.4950 0.1588
e) By definition, a node of a mode shape is simply the location of zero displacement. Determine the node locations for modes 2 to 4. Under what conditions may the node locations be important? Please give at least 2 examples. Linear interpolation of the curves of modes 2 to 4 gives the three nodes locations
3
Answer: a) Determine the third normalized eigenvector of the system Assume the third normalized eigenvector of the system to be u 3 u 31 , u 32 , u 33 , then, from the
mode Node location ( ith story) 2 3.0 3 1.7423, 3.5740 4 1.3949, 2.5321, 3.7169
If we want to excite the structure to measure the vibration and do some modal analyses, it’s important to select the proper nodes for excitation. If the point for excitation happens to be at or near a node of a mode, the specific mode will not be excited and can not be measured. A second example to illustrate the importance of the locations of nodes is related to joint two large machine parts or components together. If the points to link (assembly) the two elements together coincide with the nodes of the principle modes, the bolting force will vary little due to vibration of the structures.
Λ diag (0.1508,1.2500, 2.9341,4.4151) , ω diag (0.3883,1.1180,1.7129,2.1012)
- 0.1140 - 0.2143 u4 ] - 0.2887 - 0.3283 0.2887 0.3283 0.2887 - 0.1140 0.0000 - 0.2887 - 0.2887 0.2143 - 0.2143 0.3283 - 0.2887 0.1140
4
numerically efficient algorithms have been developed over the years specifically for the solution of a SEVP. Second, the behavior of a SEVP is much better understood, and many theorems have been formulated for a SEVP as opposed to a GEVP. Prove the orthogonality conditions of the mode shapes for the SEVP. [Hint: you may want to consider the adjoint eigenvalue problem.] Prove: If [A] is symmetric, we can prove the orthogonality conditions of the mode shapes for the SEVP. However, if [A] is nonsymmetric, we should prove the biorthogonality conditions instead. Actually the orthogonality condition can be derived easily if the biorthogonality conditions are proved. a) biorthogonality conditions For the eigenvalue problem
Simplified, we obtain
(2)
4 0 0 0
0 4 0 0
0 0 4 0
0 0 10 5 0 5 10 5 0 0 x 0 x 0 5 10 5 0 4 0 5 5 0
(3)
Solving the above GEVP with a MATLAB script program (Appendix A), we obtain the 4 modes of the 4-story building
T
(7)
Thus, the free response of the system due to the intial displacement is as expressed inn Eq.(6), where
i and ci , (i 1,2,3,4) are given by Eqs.(4) and (7) respectively.
Ku 2 Mu
we know
i
( Ku i )1 , i 1,2, ( Mu i )1
By this approach, it avoids the time-consuming computation to solve a generalized eigenvalue problem and can get the result faster and more economically. 3. In class, we considered the following symmetric generalized eigenvalue problem (GEVP) [K]u = [M]u and proved the orthogonality conditions of the mode shapes with respect to the mass and stiffness matrices. Often it is more desirable to solve a standard eigenvalue problem (SEVP) of the form [A]u = u where [A] is generally nonsymmetric. The rationale for solving the SEVP is two-fold. First, many
1 (k1 k 2 ) x1 k 2 x 2 0 m1 x 2 k 2 x1 (k 2 k 3 ) x 2 k 3 x3 0 m2 x , 3 k 3 x 2 (k 3 k 4 ) x3 k 4 x 4 0 m3 x 4 k 4 x3 k 4 x 4 0 m4 x
x ci u i sin( i t i )
i 1
4
Due to the initial displacement, we have
x c1u1 sin(1t
2
) c 2 u 2 sin( 2 t
c 4 x0
T
2
) c3 u 3 sin( 3t
(4)
U [u1
u2
u3
(5)
c) Plot of the mode shapes. The mode shapes can be plotted by the MATLAB script program (Appendix A) as well
2
d) Determine the free response of the system due to the initial displacement. Under arbitrary initial conditions, the response of the system can be expressed as
0 0 x 0 k4 k4
(1)
b) Determine the modes of vibration. Assume mi = 4000 kg and ki = 5000 N/m, for i = 1, . . . , 4. For the assumed values of mi and ki, we have, from Eq.(1),